Preemptive note: I'm looking at this problem from a 5e perspective. It's quite possible that damage immunities or resistances differ between versions, so these suggestions might not all be 100% applicable to your situation.
Necessary damage immunities/resistances
First off, modern weapons have so much firepower that you absolutely need immunity to what I'll call "physical" damage (= bludgeoning, piercing, slashing, though mostly the former two). As modern armies don't enchant their weapons or ammunition (as far as I'm aware ^^), immunity to just non-magical physical damage is sufficient.
On top of that, both fire (explosions, flamethrowers, napalm, ...) and poison attacks (chemical and biological weapons) are likely within the arsenal of modern armies, even if they're less abundant than simple bullets (which would be covered by the physical immunity I mentioned above). Therefore, I would say that whatever monster you pick needs at least some kind of resistance or immunity against one or both of these.
I'm not sure what kind of damage nukes would inflict, but fire is probably a safe bet. Other than that, probably thunder for the blast wave, and I'm not sure what the radiation would count as; poison/radiant/necrotic?
Either way, there are likely no monsters completely immune to nukes (especially the thunder part, as it's normally a pretty rare damage type), hence teleportation across vast distances or planes is a huge plus, so you don't have to tank the nuke's damage.
Disclaimer
With all that established, do note that not all of the following monsters are necessarily tamable, controllable, or otherwise subjugatable. For those that are not, you might need some house-ruling to explain why they're helping.
Furthermore, most of these creatures are at least mid-level, because with how destructive modern weapons are, CR1 creatures just won't cut it, unless they had all the right immunities - but there's just no such low-level creature.
List of options
If you're looking for mid-level creatures, then the Steel Predator is probably a good bet. It's immune to both physical and poison damage, and while it has no additional protection against fire, it does sport fairly decent armor and HP, can teleport and is overall pretty capable. Unfortunately, they're designed as assassins that normally focus on taking out individual targets (though potentially killing lots of others that are in the way) and return to their creators afterward. Thus, you'd need to get those creators on board, to make them "reusable", so to speak; or you do just use them to take out individuals, such as high-ranking generals, the president, etc.
With varying levels of strength, devils of all levels (including lords) are also an option, as they all have immunity to both poison and fire. Unfortunately, they (including lords) only have resistance against physical damage, not immunity, which means you will need a lot of them.
Regular non-lord demons are also an option, although they only have fire resistance instead of immunity.
Following up on this, a very strong all-around troop are demon lords. They are all immune to physical and poison damage and they have resistance to fire (as well as lightning, which armies would probably also be capable of inflicting somehow, although with more difficulty than fire or poison). Some of them are also capable of flight and being high-CR creatures, they all have very powerful abilities. On top of that, you can canonically control them if you get your hands on their demonic amulet (MToF, p. 26) - not that that's particularly easy.
Another option are Iron Golems, which have immunities to fire, poison, and non-adamantine physical damage. They're also easily controlled, though far from easily acquired. Weaker golems are also an option, but they lack fire immunity.
At higher levels, Liches (and to a lesser extent Demiliches) are also always a good option, due to physical and (thanks to their undead nature) poison immunities, as well as very strong and versatile abilities due to being a spellcaster. They're also pretty much impossible to kill because they can just park their phylactery on a demiplane, which armies simply have no way of accessing without magic.
Note: their stat block in the MM doesn't include the Fly spell, but I think it's safe to say that a Lich would know how to cast such a useful third-level spell. More specifically, one would expect a Lich to have all major/common wizard spells in existance in their repertoire, preparing spells as necessary to fight an Air Force.
Similar to liches, undead in general are not the worst option.
Mummy lords are strong if you can grant them immunity to fire to overrule their vulnerability to it (the Investiture of Flame spell springs to mind, but it's self-only and not on the Mummy Lord's spell list).
Flame skulls only have fire/poison immunity and piercing resistance, but can wreak havoc with fireballs.
Many other undead are also somewhat useful, such as ghostly undead, which usually have useful resistances, if not immunities.
Your ultimate option for destroying outposts, bases, and other immobile targets is likely the Tarrasque. Naturally, that's not something one can normally control, but it sports immunity to physical, fire, and poison damage, on top of being an extremely tanky and destructive CR30 monster.
Conclusion
Overall, I would say that Demon Lords, ideally with magic items to counteract their shortcomings, are your best bet. Namely, Orcus can fly and is otherwise rather strong, though he lacks a native teleportation ability to escape nukes. Other demon lords can be similarly buffed with magic items to grant them flight or other abilities. Conveniently, there's also a canonical way of controlling them, in the form of demonic amulets.
Almost equally as strong are Liches. Unfortunately, you'll probably have a hard time convincing one (much less multiple) of fighting for you, barring DM intervention.
Either of these should also be able to summon or otherwise accumulate allies in some capacity (demons or undead, respectively).
